1. Field of the Invention
The present invention relates to a method for manufacturing semiconductor devices, and a method for forming a pattern onto an exposure mask. For example, the present invention relates to a method of correcting a dimension error generated in transferring and printing a pattern of a mask to a wafer or in a subsequent wafer process, by adjusting dimensions of the pattern on the mask at each position when forming the pattern for manufacturing a semiconductor device. Particularly, the dimension error herein is based on an assumption that an error occurs when a certain figure or a figure group affects the dimension of a figure at a distant position.
2. Description of the Related Art
First, in manufacturing an LSI, an exposure mask is produced using a mask writing apparatus, etc. Next, a pattern on the exposure mask is exposed onto the resist on a silicon wafer (Si wafer) by using an optical scanner or a stepper. Then, through various steps, such as resist development and etching, a pattern of one layer is formed. An LSI is manufactured by repeating such a pattern manufacturing step some tens of times. These days an electron beam exposure apparatus is mainly used for writing or “drawing” a mask, but there are cases of using light beams. As to an apparatus for exposing a pattern of a mask, a light beam, whose wavelength is approximately 193 nm, is currently used as mentioned above. Moreover, a technique of utilizing an extreme ultraviolet light (EUV), an electron beam, or and an X-ray, whose wavelength is approximately 10 nm, has also been studied.
Regarding forming a pattern for one layer of the LSI, it goes through various steps as stated above. One of the problems of the LSI pattern formed through such various steps is that although each pattern is approximately uniform at the time of completion when seen from a local point (that is, locally, the difference with respect to the design dimension is almost uniform), the pattern dimension changes gradually when seen from the point of the whole of reticle or the whole of the inner side of a chip formed on a wafer (that is, the difference with respect to the design dimension changes gradually in the chip).
FIG. 34 shows an example of a state of a pattern dimension variation when seen from the point of the whole of the chip inside. An error shown in FIG. 34 will be called a global CD (critical dimension) error. Some examples are described below.
As the first example, there is a phenomenon called a flare generated when transferring and printing a pattern of a mask onto an Si wafer by using an optical stepper. This phenomenon is a pattern dimension variation generated by diffused reflection due to surface coarseness of a mask and/or a lens, and if there is a position where the pattern density is high, dimensions of other figures existing within several mm from the position concerned are changed by several nm to several tens of nm approximately. The phenomenon is generated not only in an optical transfer apparatus or a scanner, which is currently mainly used and utilizing an ArF (argon fluoride) excimer laser (wavelength of 193 nm) but also in an exposure apparatus (EUV stepper), which is expected to be used in the future and utilizing a wavelength of the EUV (Extreme Ultra Violet) region.
As the second example, there is a phenomenon called a loading effect generated at the time of dry etching performed for the Si wafer in the semiconductor manufacturing process. The dry etching is a step of, after forming a resist pattern by exposing and developing a resist film, etching a film in the underlying layer of the resist pattern serving as a mask, by using plasma. In this manufacturing step, the amount of a by-product created by the etching changes depending on the area of an exposed underlying layer (namely, pattern density). That is, an etching speed varies depending upon the amount of the by-product, thereby changing an etched dimension, which is the phenomenon of the loading effect. Consequently, the change of dimension depends upon the density of the pattern. For this reason, under the influence of a position where the pattern density is high, dimensions of figures existing in a region within several cm from the position are changed by several nm to several tens of nm.
In addition to the above stated cause, the loading effect also occurs due to a CMP (Chemical Mechanical Polishing) step or development of the resist, thereby changing dimensions of a pattern depending upon the density thereof.
As mentioned above, there are many phenomena by which dimensions of figures existing in a region within several mm to several cm from a position where the pattern density is high are changed by several nm to several tens of nm by under the influence of the density. For increasing the integration degree of a semiconductor integrated circuit, it is necessary to further enhance the formation accuracy in addition not only to miniaturize the figure. However, the phenomenon above mentioned hinders to raise the accuracy, which is an obstacle in advancing the high integration. The method of correcting the dimension change resulting from such a flare or a loading effect will be called global CD correction or GCD correction. The following can be cited as examples of the method of correcting the global CD. There is a method of correcting a figure dimension at each position to correct a dimension change produced in a semiconductor manufacturing step (refer to, e.g., Japanese Unexamined Patent Publication No. 2003-43661 (JP-A-2003-43661)). In this reference document (JP-A-2003-43661), a method is proposed that when calculating a correction amount, the LSI pattern is virtually divided into a plurality of mesh-like grids to use the density of a pattern inside the grid.
However, since this method is only an approximated correction as described below, it becomes difficult to further improve the high precision. This will be hereinafter described. First, formulation of a global CD error is explained. Then, the conventional method and its problem will be explained.
Now, with reference to the reference document (JP-A-2003-43661), the formulation of a global CD error will be described. An LSI pattern is divided into regions (mesh) each of which is sufficiently smaller than the distance of the influence range of a global CD error. The size of each mesh is defined to be ΔL×ΔL. When the center coordinate of the i-th mesh is defined to be xi=(xi, yi), an amount δl(xi), which is a dimension amount of a figure existing in the i-th mesh increased by a global CD error, can be expressed by the following equation (1). In this specification, it will be hereafter described as the coordinates xi=(xi, yi) or as the coordinates x=(x, y).
                              δ          ⁢                                          ⁢                      l            ⁡                          (                              x                i                            )                                      =                                            γ              d                        ⁢                                          ∑                j                            ⁢                                                g                  ⁡                                      (                                                                  x                        i                                            -                                              x                        j                        ′                                                              )                                                  ⁢                                                      ρ                    0                                    ⁡                                      (                                          x                      j                      ′                                        )                                                  ⁢                                  Δ                  L                  2                                                              +                                    γ              p                        ⁢                          f              ⁡                              (                                  x                  i                                )                                                                        (        1        )            
Alternatively, regarding the size of the mesh to be sufficiently small, if expressed by an integral expression, it can be the following equation (2).
                              δ          ⁢                                          ⁢                      l            ⁡                          (              x              )                                      =                                            γ              d                        ⁢                                          ∫                A                            ⁢                                                g                  ⁡                                      (                                          x                      -                                              x                        ′                                                              )                                                  ⁢                                                      ρ                    0                                    ⁡                                      (                                          x                      ′                                        )                                                  ⁢                                  ⅆ                                      x                    ′                                                                                +                                    γ              p                        ⁢                          f              ⁡                              (                                  x                  i                                )                                                                        (        2        )            
In the equations (1) and (2), the first term expresses a dimension change depending on the density of a pattern. In the loading effect generated in etching a wafer, this term corresponds to a dimension change depending on the density. The second term expresses a dimension change depending on only a position. In etching a wafer, this term corresponds to a dimension change generated due to unevenness of plasma used for the etching. In the equations (1) and (2), a pattern area density of the i-th mesh in the design is defined to be ρ0(xi)=ρ0(xi, yi). Influence of density of a certain position on other position is expressed as g(x)=g(x, y), and represented by standardized functions as shown in the following equations (3) and (4).
                                          ∑            j                    ⁢                      ∑                          g              ⁡                              (                                                                            x                      i                                        -                                          x                      j                                                        ,                                                            y                      i                                        -                                          y                      j                                                                      )                                                    =        1                            (        3        )                                                      ∫            A                    ⁢                                    g              ⁡                              (                x                )                                      ⁢                          ⅆ              x                                      =                                            ∫                              -                ∞                            ∞                        ⁢                                          ∫                                  -                  ∞                                ∞                            ⁢                                                g                  ⁡                                      (                                          x                      ,                      y                                        )                                                  ⁢                                  ⅆ                  x                                ⁢                                  ⅆ                  y                                                              =          1                                    (        4        )            
The global CD error depending upon the density can be expressed by a coefficient γd. Since g(x) is standardized as mentioned above, γ indicates a difference between dimensions in the cases of the pattern density being 1 and 0. The value of γ varies depending upon a process concerned. For example, in the loading effect generated in dry etching, the value of γ varies from 5 to 20 nm. That is, depending upon a pattern density, the dimension changes from 5 up to 20 nm at the maximum.
A distribution function g(x)=g(x, y) of the loading effect can be expressed by the following equation (5-1) or equation (5-2), for example. Alternatively, other suitable functions can be utilized by the apparatus used for etching etc.g(x)=(1/πσL2)exp[−(x−x′)2/σL2]  (5-1)g(x)=(γ1/πσL12)exp[−(x−x′)2/σL12]+(γ2/πσL22)exp[−(x−x′)2/σL22]  (5-2)
where γ1+γ2=1. Alternatively, to be more general, the following equations (6-1) and (6-2) are used.
                              g          ⁡                      (            x            )                          =                              ∑            k                    ⁢                                    (                                                γ                  k                                /                                  πσ                  Lk                  2                                            )                        ⁢                          exp              ⁡                              [                                                      -                                                                  (                                                  x                          -                                                      x                            ′                                                                          )                                            2                                                        /                                      σ                    Lk                    2                                                  ]                                                                        (                  6          ⁢                      -                    ⁢          1                )                                1        =                              ∑            k                    ⁢                      γ            k                                              (                  6          ⁢                      -                    ⁢          2                )            
σL, σL1, σL2, . . . herein are the indications (influence range) of the distance of the range of the global CD error, and they are approximately from several 100 μm to several cm, for example. It is supposed that the size of a mesh is sufficiently smaller than the distance of the influence range of a global CD error as mentioned above, and it may be σL/10, for example. In the cases of σL being 1 mm and 1 cm, the sizes of meshes may be set to be 0.1 mm and 1 mm respectively.
Moreover, in the equations (1) and (2), a CD error depending on the position within a mask surface is defined to be γp×f(x, y). f(x, y) is defined to be a standardized function, whose maximum value is 1. By performing such definition, γp is equivalent to a maximum error of the global CD error depending on only a position. Moreover, Σ is calculated by summing in each small region.
Next, the conventional method will be explained. According to the conventional one, for example the method described in the Patent Document 1, a dimension of a figure at a position (xi, yi) is reduced by L0(xi, yi) in the following equation (7).
                                          L            0                    ⁡                      (                          x              i                        )                          =                                            γ              d                        ⁢                                          ∑                j                            ⁢                                                g                  ⁡                                      (                                                                  x                        i                                            -                                              x                        j                        ′                                                              )                                                  ⁢                                                      ρ                    0                                    ⁡                                      (                                          x                      j                      ′                                        )                                                  ⁢                                  Δ                  L                  2                                                              +                                    γ              p                        ⁢                          f              ⁡                              (                                  x                  i                                )                                                                        (        7        )            
where ρ0(xi)=ρ0(xi, yi) is a density of the original pattern. In the conventional method, it is very simple because the dimension increased by the global CD error, that is L0(xi, yi) obtained from the equation (7), is subtracted from the dimension of the original figure. Therefore, it is not so accurate because of the following reason. That is, when calculating a dimension correction amount from the density of the original pattern and changing the dimension of the pattern by the calculated amount, since the dimension changes after being corrected, the density of the pattern corrected differs from the original pattern density. On the other hand, if it is supposed that the original pattern density is not changed and is kept, since the dimension is changed by L0(xi, yi) by using a process apparatus or a lithography apparatus for the pattern, it becomes as designed. However, as mentioned above, since the pattern density after correcting differs from that of the original pattern, the dimension after correcting cannot be the one as designed. That is, the exact dimension correction amount cannot be obtained by the above equation (7) for correcting, thereby becoming difficult to achieve an accurate global CD correction.
In addition, although it is not for correcting a global CD error, some documents are disclosed in which a solution is calculated for an equation for correcting the proximity effect generated by irradiation of an electron beam (refer to, e.g., Japanese Patent No. 3469422 and the U.S. Pat. No. 5,863,682).